Brillouin Zone of a 2D Square Lattice: Tight Binding Approximation

This Demonstration considers the Brillouin zone (BZ) -band dispersion relations for a two-dimensional square crystal lattice under the tight binding approximation. Plots are shown for the first energy -band dispersion in the first and extended Brillouin zones. You can compare the parabolic dispersion curves of free electrons in free space and the linear dispersion curves of Dirac-electrons (as in graphene at -points).

THINGS TO TRY

SNAPSHOTS

DETAILS

The energy structure of crystals depends on the interactions between orbitals in the lattice. The tight binding approximation (TB) neglects interactions between atoms separated by large distances, an approximation which greatly simplifies the analysis. In solid-state physics, the TB model calculates the electronic band structure using an approximate set of wave functions based upon superposition of orbitals located at each individual atomic site.

In a TB approximation including only first nearest neighbor s-orbital, the band structure, i.e. the electron energy dispersion in the Brillouin zone of the crystal, is given by

(1)

where is the overlap integral between s-orbitals, is the translation vector of the lattice, and is the "crystal momentum"—the quantum number for periodic systems. The band structure for a simple cubic lattice can now be readily calculated. Assuming that the bond integrals couple only to the first four nearest neighbors with position vectors equal to and , (1) gives

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(a)

(b)

(c)

Figure: (a) lattice fragment; (b) four first nearest neighbors included in transfer; (c) the firstBrillouin zone.

Setter bars let you compare the lattice electron behavior with the parabolic dispersion curves of the free electrons in free space,

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and with the linear dispersion curves of Dirac electrons (as in graphene -points),

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Snapshot 1: constant energy contours for the -band in the firstBrillouin zone of a 2D one-base simple square lattice